We present a numerical study of periodic traveling waves of the Kawahara equation. Periodic profiles are continued with a Fourier pseudospectral method, and Bloch spectra are computed to locate intervals of temporal growth. Along the sampled branch, weaker fifth-order dispersion is accompanied by greater transfer of Fourier energy to higher modes and by the appearance of unstable spectral quartets. We also compute spatial Poincaré-section diagnostics for the traveling-wave ordinary differential equation. These spatial quantities do not prove temporal instability; they are reported as separate measurements that vary over the same parameter continuation. Finally, a finite-dimensional observer is applied to Fourier channels selected from the unstable Bloch calculation. For a range of sampled gains, the corrected growth rate becomes negative while the main oscillatory profile is retained. All finite-amplitude stability, manifold, and observer conclusions are numerical and are stated at the discretizations used in the computations.
Signed RC–Schmitt oscillator arrays can realize bipartite (anti-phase two-cluster) timing patterns useful for differential signaling and quadrature clock generation, but existing event-triggered and reinforcement-learning designs for such arrays either ignore non-memoryless operating-mode changes or leave data-driven threshold tuning unconstrained, risking loss of lock. This paper closes both gaps with a single framework: a phase-reduction and structural-balance gauge argument reduces the signed, mode-switching circuit array to an unsigned Kuramoto–Sakaguchi network amenable to Lyapunov analysis; hidden semi-Markov switching, with general holding-time distributions and imperfect mode detection, models load, supply, and comparator-delay variation; and a memory-aware event-triggered rule is combined with Q-learning threshold adaptation that is provably confined to a certified admissible interval derived from the analytical locking margin. Three theorems establish practical mean frequency locking under arbitrary hidden semi-Markov switching, asymptotic bipartite phase locking for identical oscillators, and preservation of the certified locking margin throughout all Q-learning iterations, with a pinned bipartite tracking corollary for leader-following arrays. Six-node MATLAB simulations confirm chatter-free circuit operation, bipartite locking under three hidden semi-Markov modes, and a 60.7% reduction in triggering events relative to periodic coupling without violating the certified safety margin.
